# Is there a higher dimensional Fredkin gate?

The Fredkin gate is CSWAP gate. Given a control register in $$0$$ or $$1$$, the gate does nothing or swaps two target registers respectively.

Is there a higher dimensional version of this gate? I have one d-dimensional control register (this takes values from the set $$S = \{1, 2, ... , d\}$$) and $$d$$ target registers. If the control register takes on value $$i\in S$$, then I would like to swap the first target register with the $$i^{th}$$ target register.

• @MarkS aren't all swaps (controlled or otherwise) reversible trivially? Or did I misunderstand you? Nov 8 '21 at 18:31

The question seems to be about a higher-dimensional control, and not about higher-dimensional targets. For $$d=4$$, would the following circuit work?
Furthermore although this wasn't the question asked, one could instead consider another CSWAP gate that has one qubit control, and higher-dimensional qudits that are swapped conditioned on the qubit. One example also for $$d=4$$ might be below. The four bottom qubits are to be thought of as two pairs of $$4$$-dimensional qudits.